Conduit models (WK-1, YK-1)to investigate transition between explosive and effusive eruptions

T. Koyaguchi

University of Tokyo

Collaboration with:Andy Woods, Shigeo Yoshida, Helene Massol, Noriko Mitanietc.

Explosive or Effusive

Pinatubo, 1991

Unzen, 1991

Key observations

What is the minimal model to explain these extreme eruption styles?

Basic Equation for WK-1

Mass conservation

Momentum conservation

Equation of State

(1 )0

d n v

dz

1dv dPv g Fdz dz

P

nRTn

l

11

1/ 2[ { (1 )}] wv n sP n Qdz

2

2

8

0.0025

v

rF

v

r

2 ( ) /w h hQ rn K P P

gas ：

liquid ：

T constant

(Woods and Koyaguchi, 1993)

: overpressure parameter

2/10 sPnn

Obatain the relationship between P and Q

Determine exit pressure by systematically changing mass flux, Q

for different mass flux

hydrostatic

lithostatic

Nor

mal

ized

dep

th

Normalized pressure

A method to systematically investigate the features of solutions (Shooting method)

Overpressure para meter （ α ）～ 1 atm

Mass flux

Exi

t p

ress

ure

Multiple steady solutions and “negative friction”

General features of results

Geological implication of the presence of multiple solutions

出口の圧力

Mass flux

Dome eruption Explosive eruption

Dome collapseSub-sonic solution

Sonic solution

Exi

t pre

ssur

e

Atmospheric pressure + load

Increase of chamber pressure

Pressure drop due toviscous friction

Decrease in total friction due to descending fragmentation surface

Mass flux

Pressure drop due toturbulent friction

Mass flux

Mass flux

Exi

t pre

ssur

e

Mass flux

Exi

t pre

ssur

eE

xit p

ress

ure

Mass flux

Origin of the multiple solutions

Purpose of YK-1

WK-1 YK-1

Gas-lossthroughconduit wall

Gas may escape vertically.

(Yoshida and Koyaguchi, 1999)

What is the minimal model toexpress the effects of relative velocity?

Basic equations for YK-1・ 2-velocity model

・ presence of fractured turbulent flow regime

1 .l l lQ u const

.g g gQ u const

lg l1 (1 )l l l w

d dPQu g F F

dz dz

lgg g g gw

d dPQ u g F F

dz dz

gP RT

:

:

:l

g

Q

Q

Gas volume fraction

Gas mass flux (kg/m2 ・ s)

Liquid mass flux (kg/m2 ・s)

Mass conservation

Momentum conservation

Equation of state

12

8, 0lw gw

c

F u Fr

20, sgn( )4w

lw gw g g gc

F F u ur

← Poiseuille flow

←Trubulent flow

Constitutive equation describing wall friction

Before fragmentation

After fragmentation

Tentatively critical void fraction (=0.8) was chosen as afragmentation criterion.

1

lg 2lg 2

3: (1 )( ) (1 )( ) sgn( )

4

t t

g l g g l g lb b

F u u u u u ur r

0.6

0.7 0.6t

0.6 0.7

1

lg 2lg

3: (1 )( ) sgn( )

4 8

t t

Dg g l g l

b a

CF u u u u

r r

0.8

0.85 0.8t

0.8 0.85

lg 2

3: (1 )( )g l

b

F u ur

0.6 ←Stokes’ terminal veolcity

lg 2lg: (1 )( ) sgn( )

4 g g l g lc

F u u u ur

0.7 0.8 ← turbulent pipe flow

2lg

3: (1 )( ) sgn( )

8D

g g l g la

CF u u u u

r 0.85 ←high Re terminal

velocity

Constitutive equations describing gas-liquid frictionBubbly flow

Fractured turbulent flow

Gas-particle flow

Essense of YK-1

Wall friction >>friction between liquid and gas

Wall friction ～ friction between liquid and gasBoth are determined by liquid viscosity.

Wall friction ～ friction between liquid and gasBoth are determined by gas viscosity.

Determined by gas viscosityDetermined by liquid viscosity.

Whatever the details of the constitutive equations may be…

pressure

pressure

pressure

dept

h

General features of resultsve

loci

ty

Voi

d

frac

tion

End